The General Linear Model (GLM) in SPM5

1. The General Linear Model (GLM) in SPM5 Klaas Enno Stephan Wellcome Trust Centre for Neuroimaging University College London With many thanks to my colleagues in the FIL methods group, particularly Stefan Kiebel, for useful slides SPM Short Course, May 2007Wellcome Trust Centre for Neuroimaging

2. Overview of SPM Statistical parametric map (SPM) Design matrix Image time-series Kernel Realignment Smoothing General linear model Gaussian field theory Statistical inference Normalisation p <0.05 Template Parameter estimates

3. A very simple fMRI experiment One session Passive word listening versus rest 7 cycles of rest and listening Blocks of 6 scans with 7 sec TR Stimulus function Question: Is there a change in the BOLD response between listening and rest?

4. Modelling the measured data Why? Make inferences about effects of interest • Decompose data into effects and error • Form statistic using estimates of effects and error How? stimulus function effects estimate linear model statistic data error estimate

5. model specification parameter estimation hypothesis statistic Voxel-wise time series analysis Time Time BOLD signal single voxel time series SPM

6. Single voxel regression model error = + + 1 2 Time e x1 x2 BOLD signal

7. Mass-univariate analysis: voxel-wise GLM = + y • Model is specified by • Design matrix X • Assumptions about e N: number of scans p: number of regressors The design matrix embodies all available knowledge about experimentally controlled factors and potential confounds.

8. GLM assumes Gaussian “spherical” (i.i.d.) errors Examples for non-sphericity: sphericity = iid:error covariance is scalar multiple of identity matrix: Cov(e) = 2I non-identity non-identity non-independence

9. Ordinary least squares (OLS): Estimate parameters such that minimal Parameter estimation: Ordinary least squares = + X y OLS parameter estimate (assuming iid error)

10. A geometric perspective y e Design space defined by X x2 x1

11. Correlated and orthogonal regressors y x2 x2* x1 Correlated regressors = explained variance is shared between regressors When x2 is orthogonalized with regard to x1, only the parameter estimate for x1 changes, not that for x2!

12. HRF What are the problems of using the GLM for fMRI data? • BOLD responses have a delayed and dispersed form. • The BOLD signal includes substantial amounts of low-frequency noise. • The data are serially correlated (temporally autocorrelated)  this violates the assumptions of the noise model in the GLM

13. hemodynamic response function (HRF) Problem 1: Shape of BOLD responseSolution: Convolution model The response of a linear time-invariant (LTI) system is the convolution of the input with the system's response to an impulse (delta function). expected BOLD response = input function impulse response function (HRF)

14. Convolution model of the BOLD response Convolve stimulus function with a canonical hemodynamic response function (HRF):  HRF

15. Problem 2: Low-frequency noise Solution: High pass filtering S = residual forming matrix of DCT set discrete cosine transform (DCT) set

16. blue= data black = mean + low-frequency drift green= predicted response, taking into account low-frequency drift red= predicted response (with low-frequency drift explained away) High pass filtering: example

17. Problem 3: Serial correlations with 1st order autoregressive process: AR(1) autocovariance function

18. Dealing with serial correlations • Pre-colouring: impose some known autocorrelation structure on the data (filtering with matrix W) and use Satterthwaite correction for df’s. • Pre-whitening: 1. Use an enhanced noise model with hyperparameters for multiple error covariance components. 2. Use estimated autocorrelation to specify matrix W and use weighted least squares (WLS) estimation.

19. Multiple covariance components enhanced noise model Q2 Q1 V 1 + 2 = Estimation of hyperparameters  with ReML (restricted maximum likelihood).

20. Hyperparameter estimation • Assumption: error covariance pattern is identical within voxels (of same tissue class), variance is not voxel-specific estimates of , but single estimate of  for all voxels • Estimation of  in SPM proceeds by pooling over all voxels that survive an F-test at p<0.001 based on OLS estimates (“first pass” during estimation in SPM) Three hyperparameters: 1, 2, 

21. Contrasts &statistical parametric maps c = 1 0 0 0 0 0 0 0 0 0 0 Q: activation during listening ? Null hypothesis:

22. t-statistic based on ML estimates c = +1 0 0 0 0 0 0 0 0 0 0 ReML-estimate

23. Physiological confounds • head movements • arterial pulsations • breathing • eye blinks • adaptation affects, fatigue, fluctuations in concentration, etc.

24. Outlook: further challenges • correction for multiple comparisons • variability in the HRF across voxels • slice timing • limitations of frequentist statistics Bayesian analyses • GLM ignores interactions among voxels models of effective connectivity

25. Summary • Mass-univariate approach: same GLM for each voxel • GLM includes all known experimental effects and confounds • Convolution with a canonical HRF • High-pass filtering to account for low-frequency drifts • Estimation of multiple variance components (e.g. to account for serial correlations) • Parametric statistics

27. Correction for multiple comparisons • Mass-univariate approach: We apply the GLM to each of a huge number of voxels (usually > 100,000). • Threshold of p<0.05  more than 5000 voxels significant by chance! • Massive problem with multiple comparisons! • Solution: Gaussian random field theory(Will’s talk)

28. Variability in the HRF • HRF varies substantially across voxels and subjects • For example, latency can differ by ± 1 second • Solution: use multiple basis functions • See talk on er-fMRI by Christian